(-2-i).jpeg "(3+8i)(-2-i)")
Multiplying Complex Numbers: (3 + 8i)(-2 - i)
This article explores the process of multiplying two complex numbers: (3 + 8i) and (-2 - i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials in algebra.
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Expand the product:
(3 + 8i)(-2 - i) = 3(-2 - i) + 8i(-2 - i)
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Distribute:
= -6 - 3i - 16i - 8i²
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Simplify using i² = -1:
= -6 - 3i - 16i + 8
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Combine real and imaginary terms:
= (-6 + 8) + (-3 - 16)i
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Final Result:
= 2 - 19i
Therefore, the product of (3 + 8i) and (-2 - i) is 2 - 19i.
Key Points
- Distributive Property: When multiplying complex numbers, use the distributive property to expand the product.
- Simplify using i² = -1: Remember that i² is equal to -1, which is crucial for simplification.
- Combine real and imaginary terms: The final result is expressed in the standard form of a complex number (a + bi).
Application of Complex Numbers
Complex numbers have wide applications in various fields, including:
- Electrical Engineering: Analyzing AC circuits.
- Mathematics: Solving equations, exploring mathematical concepts like fractals and Fourier analysis.
- Physics: Quantum mechanics, wave phenomena.
- Computer Science: Signal processing, image processing.
By understanding the fundamentals of complex number multiplication, we can explore its applications and solve complex problems in different domains.